Fix a base field $k$. First let me loosely describe the BRST reduction in the finite-dimensional setting. For a finite-dimensional Lie algebra $\mathfrak{n}$, we can form the Clifford algebra $\mathcal{Cl}_\mathfrak{n}=\mathcal{Cl}(\mathfrak{n}\oplus\mathfrak{n}^*)$, where the bilinear form on $\mathfrak{n}\oplus\mathfrak{n}^*$ is given by the pairing between $\mathfrak{n}$ and $\mathfrak{n}^*$. Given an associative algebra $\mathcal{R}$ with a Lie algebra homomorphism $\mathfrak{n}\to\mathcal{R}$, we can make $\mathcal{Cl}\otimes\mathcal{R}$ a DG superalgebra, whose cohomology is called the BRST reduction of $\mathcal{R}$.
One wants to generalize the construction to the infinite-dimensional setting. Instead of the Clifford algebra itself, we need to pass to its Fock module (the so-called fermionic Fock space). Moreover, we need to deal with the topology of infinite-dimensional vector spaces very carefully. Beilinson et al. use the notion of Tate vector space to handle the problem.
I want to ask for references
- References on Tate's linear algebra (Tate vector space and Tate central extension);
- References on infinite-dimensional BRST reduction (better if it's written in the language of Tate's linear algebra).
